1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
|
SUBROUTINE DOP853(N,FCN,X,Y,XEND,
& RTOL,ATOL,ITOL,
& SOLOUT,IOUT,
& WORK,LWORK,IWORK,LIWORK,RPAR,IPAR,IDID)
C ----------------------------------------------------------
C NUMERICAL SOLUTION OF A SYSTEM OF FIRST 0RDER
C ORDINARY DIFFERENTIAL EQUATIONS Y'=F(X,Y).
C THIS IS AN EXPLICIT RUNGE-KUTTA METHOD OF ORDER 8(5,3)
C DUE TO DORMAND & PRINCE (WITH STEPSIZE CONTROL AND
C DENSE OUTPUT)
C
C AUTHORS: E. HAIRER AND G. WANNER
C UNIVERSITE DE GENEVE, DEPT. DE MATHEMATIQUES
C CH-1211 GENEVE 24, SWITZERLAND
C E-MAIL: Ernst.Hairer@math.unige.ch
C Gerhard.Wanner@math.unige.ch
C
C THIS CODE IS DESCRIBED IN:
C E. HAIRER, S.P. NORSETT AND G. WANNER, SOLVING ORDINARY
C DIFFERENTIAL EQUATIONS I. NONSTIFF PROBLEMS. 2ND EDITION.
C SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
C SPRINGER-VERLAG (1993)
C
C VERSION OF APRIL 25, 1996
C (latest correction of a small bug: August 8, 2005)
C
C Edited (22 Feb 2009) by J.C. Travers:
C renamed HINIT->HINIT853 to avoid name collision with dopri5
C
C INPUT PARAMETERS
C ----------------
C N DIMENSION OF THE SYSTEM
C
C FCN NAME (EXTERNAL) OF SUBROUTINE COMPUTING THE
C VALUE OF F(X,Y):
C SUBROUTINE FCN(N,X,Y,F,RPAR,IPAR)
C DOUBLE PRECISION X,Y(N),F(N)
C F(1)=... ETC.
C
C X INITIAL X-VALUE
C
C Y(N) INITIAL VALUES FOR Y
C
C XEND FINAL X-VALUE (XEND-X MAY BE POSITIVE OR NEGATIVE)
C
C RTOL,ATOL RELATIVE AND ABSOLUTE ERROR TOLERANCES. THEY
C CAN BE BOTH SCALARS OR ELSE BOTH VECTORS OF LENGTH N.
C ATOL SHOULD BE STRICTLY POSITIVE (POSSIBLY VERY SMALL)
C
C ITOL SWITCH FOR RTOL AND ATOL:
C ITOL=0: BOTH RTOL AND ATOL ARE SCALARS.
C THE CODE KEEPS, ROUGHLY, THE LOCAL ERROR OF
C Y(I) BELOW RTOL*ABS(Y(I))+ATOL
C ITOL=1: BOTH RTOL AND ATOL ARE VECTORS.
C THE CODE KEEPS THE LOCAL ERROR OF Y(I) BELOW
C RTOL(I)*ABS(Y(I))+ATOL(I).
C
C SOLOUT NAME (EXTERNAL) OF SUBROUTINE PROVIDING THE
C NUMERICAL SOLUTION DURING INTEGRATION.
C IF IOUT.GE.1, IT IS CALLED AFTER EVERY SUCCESSFUL STEP.
C SUPPLY A DUMMY SUBROUTINE IF IOUT=0.
C IT MUST HAVE THE FORM
C SUBROUTINE SOLOUT (NR,XOLD,X,Y,N,CON,ICOMP,ND,
C RPAR,IPAR,IRTRN)
C DIMENSION Y(N),CON(8*ND),ICOMP(ND)
C ....
C SOLOUT FURNISHES THE SOLUTION "Y" AT THE NR-TH
C GRID-POINT "X" (THEREBY THE INITIAL VALUE IS
C THE FIRST GRID-POINT).
C "XOLD" IS THE PRECEEDING GRID-POINT.
C "IRTRN" SERVES TO INTERRUPT THE INTEGRATION. IF IRTRN
C IS SET <0, DOP853 WILL RETURN TO THE CALLING PROGRAM.
C IF THE NUMERICAL SOLUTION IS ALTERED IN SOLOUT,
C SET IRTRN = 2
C
C ----- CONTINUOUS OUTPUT: -----
C DURING CALLS TO "SOLOUT", A CONTINUOUS SOLUTION
C FOR THE INTERVAL [XOLD,X] IS AVAILABLE THROUGH
C THE FUNCTION
C >>> CONTD8(I,S,CON,ICOMP,ND) <<<
C WHICH PROVIDES AN APPROXIMATION TO THE I-TH
C COMPONENT OF THE SOLUTION AT THE POINT S. THE VALUE
C S SHOULD LIE IN THE INTERVAL [XOLD,X].
C
C IOUT SWITCH FOR CALLING THE SUBROUTINE SOLOUT:
C IOUT=0: SUBROUTINE IS NEVER CALLED
C IOUT=1: SUBROUTINE IS USED FOR OUTPUT
C IOUT=2: DENSE OUTPUT IS PERFORMED IN SOLOUT
C (IN THIS CASE WORK(5) MUST BE SPECIFIED)
C
C WORK ARRAY OF WORKING SPACE OF LENGTH "LWORK".
C WORK(1),...,WORK(20) SERVE AS PARAMETERS FOR THE CODE.
C FOR STANDARD USE, SET THEM TO ZERO BEFORE CALLING.
C "LWORK" MUST BE AT LEAST 11*N+8*NRDENS+21
C WHERE NRDENS = IWORK(5)
C
C LWORK DECLARED LENGHT OF ARRAY "WORK".
C
C IWORK INTEGER WORKING SPACE OF LENGHT "LIWORK".
C IWORK(1),...,IWORK(20) SERVE AS PARAMETERS FOR THE CODE.
C FOR STANDARD USE, SET THEM TO ZERO BEFORE CALLING.
C "LIWORK" MUST BE AT LEAST NRDENS+21 .
C
C LIWORK DECLARED LENGHT OF ARRAY "IWORK".
C
C RPAR, IPAR REAL AND INTEGER PARAMETERS (OR PARAMETER ARRAYS) WHICH
C CAN BE USED FOR COMMUNICATION BETWEEN YOUR CALLING
C PROGRAM AND THE FCN, JAC, MAS, SOLOUT SUBROUTINES.
C
C-----------------------------------------------------------------------
C
C SOPHISTICATED SETTING OF PARAMETERS
C -----------------------------------
C SEVERAL PARAMETERS (WORK(1),...,IWORK(1),...) ALLOW
C TO ADAPT THE CODE TO THE PROBLEM AND TO THE NEEDS OF
C THE USER. FOR ZERO INPUT, THE CODE CHOOSES DEFAULT VALUES.
C
C WORK(1) UROUND, THE ROUNDING UNIT, DEFAULT 2.3D-16.
C
C WORK(2) THE SAFETY FACTOR IN STEP SIZE PREDICTION,
C DEFAULT 0.9D0.
C
C WORK(3), WORK(4) PARAMETERS FOR STEP SIZE SELECTION
C THE NEW STEP SIZE IS CHOSEN SUBJECT TO THE RESTRICTION
C WORK(3) <= HNEW/HOLD <= WORK(4)
C DEFAULT VALUES: WORK(3)=0.333D0, WORK(4)=6.D0
C
C WORK(5) IS THE "BETA" FOR STABILIZED STEP SIZE CONTROL
C (SEE SECTION IV.2). POSITIVE VALUES OF BETA ( <= 0.04 )
C MAKE THE STEP SIZE CONTROL MORE STABLE.
C NEGATIVE WORK(5) PROVOKE BETA=0.
C DEFAULT 0.0D0.
C
C WORK(6) MAXIMAL STEP SIZE, DEFAULT XEND-X.
C
C WORK(7) INITIAL STEP SIZE, FOR WORK(7)=0.D0 AN INITIAL GUESS
C IS COMPUTED WITH HELP OF THE FUNCTION HINIT
C
C IWORK(1) THIS IS THE MAXIMAL NUMBER OF ALLOWED STEPS.
C THE DEFAULT VALUE (FOR IWORK(1)=0) IS 100000.
C
C IWORK(2) SWITCH FOR THE CHOICE OF THE COEFFICIENTS
C IF IWORK(2).EQ.1 METHOD DOP853 OF DORMAND AND PRINCE
C (SECTION II.6).
C THE DEFAULT VALUE (FOR IWORK(2)=0) IS IWORK(2)=1.
C
C IWORK(3) SWITCH FOR PRINTING ERROR MESSAGES
C IF IWORK(3).LT.0 NO MESSAGES ARE BEING PRINTED
C IF IWORK(3).GT.0 MESSAGES ARE PRINTED WITH
C WRITE (IWORK(3),*) ...
C DEFAULT VALUE (FOR IWORK(3)=0) IS IWORK(3)=6
C
C IWORK(4) TEST FOR STIFFNESS IS ACTIVATED AFTER STEP NUMBER
C J*IWORK(4) (J INTEGER), PROVIDED IWORK(4).GT.0.
C FOR NEGATIVE IWORK(4) THE STIFFNESS TEST IS
C NEVER ACTIVATED; DEFAULT VALUE IS IWORK(4)=1000
C
C IWORK(5) = NRDENS = NUMBER OF COMPONENTS, FOR WHICH DENSE OUTPUT
C IS REQUIRED; DEFAULT VALUE IS IWORK(5)=0;
C FOR 0 < NRDENS < N THE COMPONENTS (FOR WHICH DENSE
C OUTPUT IS REQUIRED) HAVE TO BE SPECIFIED IN
C IWORK(21),...,IWORK(NRDENS+20);
C FOR NRDENS=N THIS IS DONE BY THE CODE.
C
C----------------------------------------------------------------------
C
C OUTPUT PARAMETERS
C -----------------
C X X-VALUE FOR WHICH THE SOLUTION HAS BEEN COMPUTED
C (AFTER SUCCESSFUL RETURN X=XEND).
C
C Y(N) NUMERICAL SOLUTION AT X
C
C H PREDICTED STEP SIZE OF THE LAST ACCEPTED STEP
C
C IDID REPORTS ON SUCCESSFULNESS UPON RETURN:
C IDID= 1 COMPUTATION SUCCESSFUL,
C IDID= 2 COMPUT. SUCCESSFUL (INTERRUPTED BY SOLOUT)
C IDID=-1 INPUT IS NOT CONSISTENT,
C IDID=-2 LARGER NMAX IS NEEDED,
C IDID=-3 STEP SIZE BECOMES TOO SMALL.
C IDID=-4 PROBLEM IS PROBABLY STIFF (INTERRUPTED).
C
C IWORK(17) NFCN NUMBER OF FUNCTION EVALUATIONS
C IWORK(18) NSTEP NUMBER OF COMPUTED STEPS
C IWORK(19) NACCPT NUMBER OF ACCEPTED STEPS
C IWORK(20) NREJCT NUMBER OF REJECTED STEPS (DUE TO ERROR TEST),
C (STEP REJECTIONS IN THE FIRST STEP ARE NOT COUNTED)
C-----------------------------------------------------------------------
C *** *** *** *** *** *** *** *** *** *** *** *** ***
C DECLARATIONS
C *** *** *** *** *** *** *** *** *** *** *** *** ***
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION Y(N),ATOL(*),RTOL(*),WORK(LWORK),IWORK(LIWORK)
DIMENSION RPAR(*),IPAR(*)
LOGICAL ARRET
EXTERNAL FCN,SOLOUT
C *** *** *** *** *** *** ***
C SETTING THE PARAMETERS
C *** *** *** *** *** *** ***
NFCN=0
NSTEP=0
NACCPT=0
NREJCT=0
ARRET=.FALSE.
C -------- IPRINT FOR MONITORING THE PRINTING
IF(IWORK(3).EQ.0)THEN
IPRINT=6
ELSE
IPRINT=IWORK(3)
END IF
C -------- NMAX , THE MAXIMAL NUMBER OF STEPS -----
IF(IWORK(1).EQ.0)THEN
NMAX=100000
ELSE
NMAX=IWORK(1)
IF(NMAX.LE.0)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' WRONG INPUT IWORK(1)=',IWORK(1)
ARRET=.TRUE.
END IF
END IF
C -------- METH COEFFICIENTS OF THE METHOD
IF(IWORK(2).EQ.0)THEN
METH=1
ELSE
METH=IWORK(2)
IF(METH.LE.0.OR.METH.GE.4)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT IWORK(2)=',IWORK(2)
ARRET=.TRUE.
END IF
END IF
C -------- NSTIFF PARAMETER FOR STIFFNESS DETECTION
NSTIFF=IWORK(4)
IF (NSTIFF.EQ.0) NSTIFF=1000
IF (NSTIFF.LT.0) NSTIFF=NMAX+10
C -------- NRDENS NUMBER OF DENSE OUTPUT COMPONENTS
NRDENS=IWORK(5)
IF(NRDENS.LT.0.OR.NRDENS.GT.N)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT IWORK(5)=',IWORK(5)
ARRET=.TRUE.
ELSE
IF(NRDENS.GT.0.AND.IOUT.LT.2)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' WARNING: PUT IOUT=2 FOR DENSE OUTPUT '
END IF
IF (NRDENS.EQ.N) THEN
DO I=1,NRDENS
IWORK(I+20)=I
END DO
END IF
END IF
C -------- UROUND SMALLEST NUMBER SATISFYING 1.D0+UROUND>1.D0
IF(WORK(1).EQ.0.D0)THEN
UROUND=2.3D-16
ELSE
UROUND=WORK(1)
IF(UROUND.LE.1.D-35.OR.UROUND.GE.1.D0)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' WHICH MACHINE DO YOU HAVE? YOUR UROUND WAS:',WORK(1)
ARRET=.TRUE.
END IF
END IF
C ------- SAFETY FACTOR -------------
IF(WORK(2).EQ.0.D0)THEN
SAFE=0.9D0
ELSE
SAFE=WORK(2)
IF(SAFE.GE.1.D0.OR.SAFE.LE.1.D-4)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT FOR SAFETY FACTOR WORK(2)=',WORK(2)
ARRET=.TRUE.
END IF
END IF
C ------- FAC1,FAC2 PARAMETERS FOR STEP SIZE SELECTION
IF(WORK(3).EQ.0.D0)THEN
FAC1=0.333D0
ELSE
FAC1=WORK(3)
END IF
IF(WORK(4).EQ.0.D0)THEN
FAC2=6.D0
ELSE
FAC2=WORK(4)
END IF
C --------- BETA FOR STEP CONTROL STABILIZATION -----------
IF(WORK(5).EQ.0.D0)THEN
BETA=0.0D0
ELSE
IF(WORK(5).LT.0.D0)THEN
BETA=0.D0
ELSE
BETA=WORK(5)
IF(BETA.GT.0.2D0)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' CURIOUS INPUT FOR BETA: WORK(5)=',WORK(5)
ARRET=.TRUE.
END IF
END IF
END IF
C -------- MAXIMAL STEP SIZE
IF(WORK(6).EQ.0.D0)THEN
HMAX=XEND-X
ELSE
HMAX=WORK(6)
END IF
C -------- INITIAL STEP SIZE
H=WORK(7)
C ------- PREPARE THE ENTRY-POINTS FOR THE ARRAYS IN WORK -----
IEK1=21
IEK2=IEK1+N
IEK3=IEK2+N
IEK4=IEK3+N
IEK5=IEK4+N
IEK6=IEK5+N
IEK7=IEK6+N
IEK8=IEK7+N
IEK9=IEK8+N
IEK10=IEK9+N
IEY1=IEK10+N
IECO=IEY1+N
C ------ TOTAL STORAGE REQUIREMENT -----------
ISTORE=IECO+8*NRDENS-1
IF(ISTORE.GT.LWORK)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' INSUFFICIENT STORAGE FOR WORK, MIN. LWORK=',ISTORE
ARRET=.TRUE.
END IF
ICOMP=21
ISTORE=ICOMP+NRDENS-1
IF(ISTORE.GT.LIWORK)THEN
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' INSUFFICIENT STORAGE FOR IWORK, MIN. LIWORK=',ISTORE
ARRET=.TRUE.
END IF
C -------- WHEN A FAIL HAS OCCURED, WE RETURN WITH IDID=-1
IF (ARRET) THEN
IDID=-1
RETURN
END IF
C -------- CALL TO CORE INTEGRATOR ------------
CALL DP86CO(N,FCN,X,Y,XEND,HMAX,H,RTOL,ATOL,ITOL,IPRINT,
& SOLOUT,IOUT,IDID,NMAX,UROUND,METH,NSTIFF,SAFE,BETA,FAC1,FAC2,
& WORK(IEK1),WORK(IEK2),WORK(IEK3),WORK(IEK4),WORK(IEK5),
& WORK(IEK6),WORK(IEK7),WORK(IEK8),WORK(IEK9),WORK(IEK10),
& WORK(IEY1),WORK(IECO),IWORK(ICOMP),NRDENS,RPAR,IPAR,
& NFCN,NSTEP,NACCPT,NREJCT)
WORK(7)=H
IWORK(17)=NFCN
IWORK(18)=NSTEP
IWORK(19)=NACCPT
IWORK(20)=NREJCT
C ----------- RETURN -----------
RETURN
END
C
C
C
C ----- ... AND HERE IS THE CORE INTEGRATOR ----------
C
SUBROUTINE DP86CO(N,FCN,X,Y,XEND,HMAX,H,RTOL,ATOL,ITOL,IPRINT,
& SOLOUT,IOUT,IDID,NMAX,UROUND,METH,NSTIFF,SAFE,BETA,FAC1,FAC2,
& K1,K2,K3,K4,K5,K6,K7,K8,K9,K10,Y1,CONT,ICOMP,NRD,RPAR,IPAR,
& NFCN,NSTEP,NACCPT,NREJCT)
C ----------------------------------------------------------
C CORE INTEGRATOR FOR DOP853
C PARAMETERS SAME AS IN DOP853 WITH WORKSPACE ADDED
C ----------------------------------------------------------
C DECLARATIONS
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
parameter (
& c2 = 0.526001519587677318785587544488D-01,
& c3 = 0.789002279381515978178381316732D-01,
& c4 = 0.118350341907227396726757197510D+00,
& c5 = 0.281649658092772603273242802490D+00,
& c6 = 0.333333333333333333333333333333D+00,
& c7 = 0.25D+00,
& c8 = 0.307692307692307692307692307692D+00,
& c9 = 0.651282051282051282051282051282D+00,
& c10 = 0.6D+00,
& c11 = 0.857142857142857142857142857142D+00,
& c14 = 0.1D+00,
& c15 = 0.2D+00,
& c16 = 0.777777777777777777777777777778D+00)
parameter (
& b1 = 5.42937341165687622380535766363D-2,
& b6 = 4.45031289275240888144113950566D0,
& b7 = 1.89151789931450038304281599044D0,
& b8 = -5.8012039600105847814672114227D0,
& b9 = 3.1116436695781989440891606237D-1,
& b10 = -1.52160949662516078556178806805D-1,
& b11 = 2.01365400804030348374776537501D-1,
& b12 = 4.47106157277725905176885569043D-2)
parameter (
& bhh1 = 0.244094488188976377952755905512D+00,
& bhh2 = 0.733846688281611857341361741547D+00,
& bhh3 = 0.220588235294117647058823529412D-01)
parameter (
& er 1 = 0.1312004499419488073250102996D-01,
& er 6 = -0.1225156446376204440720569753D+01,
& er 7 = -0.4957589496572501915214079952D+00,
& er 8 = 0.1664377182454986536961530415D+01,
& er 9 = -0.3503288487499736816886487290D+00,
& er10 = 0.3341791187130174790297318841D+00,
& er11 = 0.8192320648511571246570742613D-01,
& er12 = -0.2235530786388629525884427845D-01)
parameter (
& a21 = 5.26001519587677318785587544488D-2,
& a31 = 1.97250569845378994544595329183D-2,
& a32 = 5.91751709536136983633785987549D-2,
& a41 = 2.95875854768068491816892993775D-2,
& a43 = 8.87627564304205475450678981324D-2,
& a51 = 2.41365134159266685502369798665D-1,
& a53 = -8.84549479328286085344864962717D-1,
& a54 = 9.24834003261792003115737966543D-1,
& a61 = 3.7037037037037037037037037037D-2,
& a64 = 1.70828608729473871279604482173D-1,
& a65 = 1.25467687566822425016691814123D-1,
& a71 = 3.7109375D-2,
& a74 = 1.70252211019544039314978060272D-1,
& a75 = 6.02165389804559606850219397283D-2,
& a76 = -1.7578125D-2)
parameter (
& a81 = 3.70920001185047927108779319836D-2,
& a84 = 1.70383925712239993810214054705D-1,
& a85 = 1.07262030446373284651809199168D-1,
& a86 = -1.53194377486244017527936158236D-2,
& a87 = 8.27378916381402288758473766002D-3,
& a91 = 6.24110958716075717114429577812D-1,
& a94 = -3.36089262944694129406857109825D0,
& a95 = -8.68219346841726006818189891453D-1,
& a96 = 2.75920996994467083049415600797D1,
& a97 = 2.01540675504778934086186788979D1,
& a98 = -4.34898841810699588477366255144D1,
& a101 = 4.77662536438264365890433908527D-1,
& a104 = -2.48811461997166764192642586468D0,
& a105 = -5.90290826836842996371446475743D-1,
& a106 = 2.12300514481811942347288949897D1,
& a107 = 1.52792336328824235832596922938D1,
& a108 = -3.32882109689848629194453265587D1,
& a109 = -2.03312017085086261358222928593D-2)
parameter (
& a111 = -9.3714243008598732571704021658D-1,
& a114 = 5.18637242884406370830023853209D0,
& a115 = 1.09143734899672957818500254654D0,
& a116 = -8.14978701074692612513997267357D0,
& a117 = -1.85200656599969598641566180701D1,
& a118 = 2.27394870993505042818970056734D1,
& a119 = 2.49360555267965238987089396762D0,
& a1110 = -3.0467644718982195003823669022D0,
& a121 = 2.27331014751653820792359768449D0,
& a124 = -1.05344954667372501984066689879D1,
& a125 = -2.00087205822486249909675718444D0,
& a126 = -1.79589318631187989172765950534D1,
& a127 = 2.79488845294199600508499808837D1,
& a128 = -2.85899827713502369474065508674D0,
& a129 = -8.87285693353062954433549289258D0,
& a1210 = 1.23605671757943030647266201528D1,
& a1211 = 6.43392746015763530355970484046D-1)
parameter (
& a141 = 5.61675022830479523392909219681D-2,
& a147 = 2.53500210216624811088794765333D-1,
& a148 = -2.46239037470802489917441475441D-1,
& a149 = -1.24191423263816360469010140626D-1,
& a1410 = 1.5329179827876569731206322685D-1,
& a1411 = 8.20105229563468988491666602057D-3,
& a1412 = 7.56789766054569976138603589584D-3,
& a1413 = -8.298D-3)
parameter (
& a151 = 3.18346481635021405060768473261D-2,
& a156 = 2.83009096723667755288322961402D-2,
& a157 = 5.35419883074385676223797384372D-2,
& a158 = -5.49237485713909884646569340306D-2,
& a1511 = -1.08347328697249322858509316994D-4,
& a1512 = 3.82571090835658412954920192323D-4,
& a1513 = -3.40465008687404560802977114492D-4,
& a1514 = 1.41312443674632500278074618366D-1,
& a161 = -4.28896301583791923408573538692D-1,
& a166 = -4.69762141536116384314449447206D0,
& a167 = 7.68342119606259904184240953878D0,
& a168 = 4.06898981839711007970213554331D0,
& a169 = 3.56727187455281109270669543021D-1,
& a1613 = -1.39902416515901462129418009734D-3,
& a1614 = 2.9475147891527723389556272149D0,
& a1615 = -9.15095847217987001081870187138D0)
parameter (
& d41 = -0.84289382761090128651353491142D+01,
& d46 = 0.56671495351937776962531783590D+00,
& d47 = -0.30689499459498916912797304727D+01,
& d48 = 0.23846676565120698287728149680D+01,
& d49 = 0.21170345824450282767155149946D+01,
& d410 = -0.87139158377797299206789907490D+00,
& d411 = 0.22404374302607882758541771650D+01,
& d412 = 0.63157877876946881815570249290D+00,
& d413 = -0.88990336451333310820698117400D-01,
& d414 = 0.18148505520854727256656404962D+02,
& d415 = -0.91946323924783554000451984436D+01,
& d416 = -0.44360363875948939664310572000D+01)
parameter (
& d51 = 0.10427508642579134603413151009D+02,
& d56 = 0.24228349177525818288430175319D+03,
& d57 = 0.16520045171727028198505394887D+03,
& d58 = -0.37454675472269020279518312152D+03,
& d59 = -0.22113666853125306036270938578D+02,
& d510 = 0.77334326684722638389603898808D+01,
& d511 = -0.30674084731089398182061213626D+02,
& d512 = -0.93321305264302278729567221706D+01,
& d513 = 0.15697238121770843886131091075D+02,
& d514 = -0.31139403219565177677282850411D+02,
& d515 = -0.93529243588444783865713862664D+01,
& d516 = 0.35816841486394083752465898540D+02)
parameter (
& d61 = 0.19985053242002433820987653617D+02,
& d66 = -0.38703730874935176555105901742D+03,
& d67 = -0.18917813819516756882830838328D+03,
& d68 = 0.52780815920542364900561016686D+03,
& d69 = -0.11573902539959630126141871134D+02,
& d610 = 0.68812326946963000169666922661D+01,
& d611 = -0.10006050966910838403183860980D+01,
& d612 = 0.77771377980534432092869265740D+00,
& d613 = -0.27782057523535084065932004339D+01,
& d614 = -0.60196695231264120758267380846D+02,
& d615 = 0.84320405506677161018159903784D+02,
& d616 = 0.11992291136182789328035130030D+02)
parameter (
& d71 = -0.25693933462703749003312586129D+02,
& d76 = -0.15418974869023643374053993627D+03,
& d77 = -0.23152937917604549567536039109D+03,
& d78 = 0.35763911791061412378285349910D+03,
& d79 = 0.93405324183624310003907691704D+02,
& d710 = -0.37458323136451633156875139351D+02,
& d711 = 0.10409964950896230045147246184D+03,
& d712 = 0.29840293426660503123344363579D+02,
& d713 = -0.43533456590011143754432175058D+02,
& d714 = 0.96324553959188282948394950600D+02,
& d715 = -0.39177261675615439165231486172D+02,
& d716 = -0.14972683625798562581422125276D+03)
DOUBLE PRECISION Y(N),Y1(N),K1(N),K2(N),K3(N),K4(N),K5(N),K6(N)
DOUBLE PRECISION K7(N),K8(N),K9(N),K10(N),ATOL(*),RTOL(*)
DIMENSION CONT(8*NRD),ICOMP(NRD),RPAR(*),IPAR(*)
LOGICAL REJECT,LAST
EXTERNAL FCN
COMMON /CONDO8/XOLD,HOUT
C *** *** *** *** *** *** ***
C INITIALISATIONS
C *** *** *** *** *** *** ***
FACOLD=1.D-4
EXPO1=1.d0/8.d0-BETA*0.2D0
FACC1=1.D0/FAC1
FACC2=1.D0/FAC2
POSNEG=SIGN(1.D0,XEND-X)
C --- INITIAL PREPARATIONS
ATOLI=ATOL(1)
RTOLI=RTOL(1)
LAST=.FALSE.
HLAMB=0.D0
IASTI=0
CALL FCN(N,X,Y,K1,RPAR,IPAR)
HMAX=ABS(HMAX)
IORD=8
IF (H.EQ.0.D0) H=HINIT853(N,FCN,X,Y,XEND,POSNEG,K1,K2,K3,IORD,
& HMAX,ATOL,RTOL,ITOL,RPAR,IPAR)
NFCN=NFCN+2
REJECT=.FALSE.
XOLD=X
IF (IOUT.GE.1) THEN
IRTRN=1
HOUT=1.D0
CALL SOLOUT(NACCPT+1,XOLD,X,Y,N,CONT,ICOMP,NRD,
& RPAR,IPAR,IRTRN)
IF (IRTRN.LT.0) GOTO 79
END IF
C --- BASIC INTEGRATION STEP
1 CONTINUE
IF (NSTEP.GT.NMAX) GOTO 78
IF (0.1D0*ABS(H).LE.ABS(X)*UROUND)GOTO 77
IF ((X+1.01D0*H-XEND)*POSNEG.GT.0.D0) THEN
H=XEND-X
LAST=.TRUE.
END IF
NSTEP=NSTEP+1
C --- THE TWELVE STAGES
IF (IRTRN.GE.2) THEN
CALL FCN(N,X,Y,K1,RPAR,IPAR)
END IF
DO 22 I=1,N
22 Y1(I)=Y(I)+H*A21*K1(I)
CALL FCN(N,X+C2*H,Y1,K2,RPAR,IPAR)
DO 23 I=1,N
23 Y1(I)=Y(I)+H*(A31*K1(I)+A32*K2(I))
CALL FCN(N,X+C3*H,Y1,K3,RPAR,IPAR)
DO 24 I=1,N
24 Y1(I)=Y(I)+H*(A41*K1(I)+A43*K3(I))
CALL FCN(N,X+C4*H,Y1,K4,RPAR,IPAR)
DO 25 I=1,N
25 Y1(I)=Y(I)+H*(A51*K1(I)+A53*K3(I)+A54*K4(I))
CALL FCN(N,X+C5*H,Y1,K5,RPAR,IPAR)
DO 26 I=1,N
26 Y1(I)=Y(I)+H*(A61*K1(I)+A64*K4(I)+A65*K5(I))
CALL FCN(N,X+C6*H,Y1,K6,RPAR,IPAR)
DO 27 I=1,N
27 Y1(I)=Y(I)+H*(A71*K1(I)+A74*K4(I)+A75*K5(I)+A76*K6(I))
CALL FCN(N,X+C7*H,Y1,K7,RPAR,IPAR)
DO 28 I=1,N
28 Y1(I)=Y(I)+H*(A81*K1(I)+A84*K4(I)+A85*K5(I)+A86*K6(I)+A87*K7(I))
CALL FCN(N,X+C8*H,Y1,K8,RPAR,IPAR)
DO 29 I=1,N
29 Y1(I)=Y(I)+H*(A91*K1(I)+A94*K4(I)+A95*K5(I)+A96*K6(I)+A97*K7(I)
& +A98*K8(I))
CALL FCN(N,X+C9*H,Y1,K9,RPAR,IPAR)
DO 30 I=1,N
30 Y1(I)=Y(I)+H*(A101*K1(I)+A104*K4(I)+A105*K5(I)+A106*K6(I)
& +A107*K7(I)+A108*K8(I)+A109*K9(I))
CALL FCN(N,X+C10*H,Y1,K10,RPAR,IPAR)
DO 31 I=1,N
31 Y1(I)=Y(I)+H*(A111*K1(I)+A114*K4(I)+A115*K5(I)+A116*K6(I)
& +A117*K7(I)+A118*K8(I)+A119*K9(I)+A1110*K10(I))
CALL FCN(N,X+C11*H,Y1,K2,RPAR,IPAR)
XPH=X+H
DO 32 I=1,N
32 Y1(I)=Y(I)+H*(A121*K1(I)+A124*K4(I)+A125*K5(I)+A126*K6(I)
& +A127*K7(I)+A128*K8(I)+A129*K9(I)+A1210*K10(I)+A1211*K2(I))
CALL FCN(N,XPH,Y1,K3,RPAR,IPAR)
NFCN=NFCN+11
DO 35 I=1,N
K4(I)=B1*K1(I)+B6*K6(I)+B7*K7(I)+B8*K8(I)+B9*K9(I)
& +B10*K10(I)+B11*K2(I)+B12*K3(I)
35 K5(I)=Y(I)+H*K4(I)
C --- ERROR ESTIMATION
ERR=0.D0
ERR2=0.D0
IF (ITOL.EQ.0) THEN
DO 41 I=1,N
SK=ATOLI+RTOLI*MAX(ABS(Y(I)),ABS(K5(I)))
ERRI=K4(I)-BHH1*K1(I)-BHH2*K9(I)-BHH3*K3(I)
ERR2=ERR2+(ERRI/SK)**2
ERRI=ER1*K1(I)+ER6*K6(I)+ER7*K7(I)+ER8*K8(I)+ER9*K9(I)
& +ER10*K10(I)+ER11*K2(I)+ER12*K3(I)
41 ERR=ERR+(ERRI/SK)**2
ELSE
DO 42 I=1,N
SK=ATOL(I)+RTOL(I)*MAX(ABS(Y(I)),ABS(K5(I)))
ERRI=K4(I)-BHH1*K1(I)-BHH2*K9(I)-BHH3*K3(I)
ERR2=ERR2+(ERRI/SK)**2
ERRI=ER1*K1(I)+ER6*K6(I)+ER7*K7(I)+ER8*K8(I)+ER9*K9(I)
& +ER10*K10(I)+ER11*K2(I)+ER12*K3(I)
42 ERR=ERR+(ERRI/SK)**2
END IF
DENO=ERR+0.01D0*ERR2
IF (DENO.LE.0.D0) DENO=1.D0
ERR=ABS(H)*ERR*SQRT(1.D0/(N*DENO))
C --- COMPUTATION OF HNEW
FAC11=ERR**EXPO1
C --- LUND-STABILIZATION
FAC=FAC11/FACOLD**BETA
C --- WE REQUIRE FAC1 <= HNEW/H <= FAC2
FAC=MAX(FACC2,MIN(FACC1,FAC/SAFE))
HNEW=H/FAC
IF(ERR.LE.1.D0)THEN
C --- STEP IS ACCEPTED
FACOLD=MAX(ERR,1.0D-4)
NACCPT=NACCPT+1
CALL FCN(N,XPH,K5,K4,RPAR,IPAR)
NFCN=NFCN+1
C ------- STIFFNESS DETECTION
IF (MOD(NACCPT,NSTIFF).EQ.0.OR.IASTI.GT.0) THEN
STNUM=0.D0
STDEN=0.D0
DO 64 I=1,N
STNUM=STNUM+(K4(I)-K3(I))**2
STDEN=STDEN+(K5(I)-Y1(I))**2
64 CONTINUE
IF (STDEN.GT.0.D0) HLAMB=ABS(H)*SQRT(STNUM/STDEN)
IF (HLAMB.GT.6.1D0) THEN
NONSTI=0
IASTI=IASTI+1
IF (IASTI.EQ.15) THEN
IF (IPRINT.GT.0) WRITE (IPRINT,*)
& ' THE PROBLEM SEEMS TO BECOME STIFF AT X = ',X
IF (IPRINT.LE.0) GOTO 76
END IF
ELSE
NONSTI=NONSTI+1
IF (NONSTI.EQ.6) IASTI=0
END IF
END IF
C ------- FINAL PREPARATION FOR DENSE OUTPUT
IF (IOUT.GE.2) THEN
C ---- SAVE THE FIRST FUNCTION EVALUATIONS
DO 62 J=1,NRD
I=ICOMP(J)
CONT(J)=Y(I)
YDIFF=K5(I)-Y(I)
CONT(J+NRD)=YDIFF
BSPL=H*K1(I)-YDIFF
CONT(J+NRD*2)=BSPL
CONT(J+NRD*3)=YDIFF-H*K4(I)-BSPL
CONT(J+NRD*4)=D41*K1(I)+D46*K6(I)+D47*K7(I)+D48*K8(I)
& +D49*K9(I)+D410*K10(I)+D411*K2(I)+D412*K3(I)
CONT(J+NRD*5)=D51*K1(I)+D56*K6(I)+D57*K7(I)+D58*K8(I)
& +D59*K9(I)+D510*K10(I)+D511*K2(I)+D512*K3(I)
CONT(J+NRD*6)=D61*K1(I)+D66*K6(I)+D67*K7(I)+D68*K8(I)
& +D69*K9(I)+D610*K10(I)+D611*K2(I)+D612*K3(I)
CONT(J+NRD*7)=D71*K1(I)+D76*K6(I)+D77*K7(I)+D78*K8(I)
& +D79*K9(I)+D710*K10(I)+D711*K2(I)+D712*K3(I)
62 CONTINUE
C --- THE NEXT THREE FUNCTION EVALUATIONS
DO 51 I=1,N
51 Y1(I)=Y(I)+H*(A141*K1(I)+A147*K7(I)+A148*K8(I)
& +A149*K9(I)+A1410*K10(I)+A1411*K2(I)+A1412*K3(I)
& +A1413*K4(I))
CALL FCN(N,X+C14*H,Y1,K10,RPAR,IPAR)
DO 52 I=1,N
52 Y1(I)=Y(I)+H*(A151*K1(I)+A156*K6(I)+A157*K7(I)
& +A158*K8(I)+A1511*K2(I)+A1512*K3(I)+A1513*K4(I)
& +A1514*K10(I))
CALL FCN(N,X+C15*H,Y1,K2,RPAR,IPAR)
DO 53 I=1,N
53 Y1(I)=Y(I)+H*(A161*K1(I)+A166*K6(I)+A167*K7(I)
& +A168*K8(I)+A169*K9(I)+A1613*K4(I)+A1614*K10(I)
& +A1615*K2(I))
CALL FCN(N,X+C16*H,Y1,K3,RPAR,IPAR)
NFCN=NFCN+3
C --- FINAL PREPARATION
DO 63 J=1,NRD
I=ICOMP(J)
CONT(J+NRD*4)=H*(CONT(J+NRD*4)+D413*K4(I)+D414*K10(I)
& +D415*K2(I)+D416*K3(I))
CONT(J+NRD*5)=H*(CONT(J+NRD*5)+D513*K4(I)+D514*K10(I)
& +D515*K2(I)+D516*K3(I))
CONT(J+NRD*6)=H*(CONT(J+NRD*6)+D613*K4(I)+D614*K10(I)
& +D615*K2(I)+D616*K3(I))
CONT(J+NRD*7)=H*(CONT(J+NRD*7)+D713*K4(I)+D714*K10(I)
& +D715*K2(I)+D716*K3(I))
63 CONTINUE
HOUT=H
END IF
DO 67 I=1,N
K1(I)=K4(I)
67 Y(I)=K5(I)
XOLD=X
X=XPH
IF (IOUT.GE.1) THEN
CALL SOLOUT(NACCPT+1,XOLD,X,Y,N,CONT,ICOMP,NRD,
& RPAR,IPAR,IRTRN)
IF (IRTRN.LT.0) GOTO 79
END IF
C ------- NORMAL EXIT
IF (LAST) THEN
H=HNEW
IDID=1
RETURN
END IF
IF(ABS(HNEW).GT.HMAX)HNEW=POSNEG*HMAX
IF(REJECT)HNEW=POSNEG*MIN(ABS(HNEW),ABS(H))
REJECT=.FALSE.
ELSE
C --- STEP IS REJECTED
HNEW=H/MIN(FACC1,FAC11/SAFE)
REJECT=.TRUE.
IF(NACCPT.GE.1)NREJCT=NREJCT+1
LAST=.FALSE.
END IF
H=HNEW
GOTO 1
C --- FAIL EXIT
76 CONTINUE
IDID=-4
RETURN
77 CONTINUE
IF (IPRINT.GT.0) WRITE(IPRINT,979)X
IF (IPRINT.GT.0) WRITE(IPRINT,*)' STEP SIZE TOO SMALL, H=',H
IDID=-3
RETURN
78 CONTINUE
IF (IPRINT.GT.0) WRITE(IPRINT,979)X
IF (IPRINT.GT.0) WRITE(IPRINT,*)
& ' MORE THAN NMAX =',NMAX,'STEPS ARE NEEDED'
IDID=-2
RETURN
79 CONTINUE
IF (IPRINT.GT.0) WRITE(IPRINT,979)X
979 FORMAT(' EXIT OF DOP853 AT X=',E18.4)
IDID=2
RETURN
END
C
FUNCTION HINIT853(N,FCN,X,Y,XEND,POSNEG,F0,F1,Y1,IORD,
& HMAX,ATOL,RTOL,ITOL,RPAR,IPAR)
C ----------------------------------------------------------
C ---- COMPUTATION OF AN INITIAL STEP SIZE GUESS
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION Y(N),Y1(N),F0(N),F1(N),ATOL(*),RTOL(*)
DIMENSION RPAR(*),IPAR(*)
C ---- COMPUTE A FIRST GUESS FOR EXPLICIT EULER AS
C ---- H = 0.01 * NORM (Y0) / NORM (F0)
C ---- THE INCREMENT FOR EXPLICIT EULER IS SMALL
C ---- COMPARED TO THE SOLUTION
DNF=0.0D0
DNY=0.0D0
ATOLI=ATOL(1)
RTOLI=RTOL(1)
IF (ITOL.EQ.0) THEN
DO 10 I=1,N
SK=ATOLI+RTOLI*ABS(Y(I))
DNF=DNF+(F0(I)/SK)**2
10 DNY=DNY+(Y(I)/SK)**2
ELSE
DO 11 I=1,N
SK=ATOL(I)+RTOL(I)*ABS(Y(I))
DNF=DNF+(F0(I)/SK)**2
11 DNY=DNY+(Y(I)/SK)**2
END IF
IF (DNF.LE.1.D-10.OR.DNY.LE.1.D-10) THEN
H=1.0D-6
ELSE
H=SQRT(DNY/DNF)*0.01D0
END IF
H=MIN(H,HMAX)
H=SIGN(H,POSNEG)
C ---- PERFORM AN EXPLICIT EULER STEP
DO 12 I=1,N
12 Y1(I)=Y(I)+H*F0(I)
CALL FCN(N,X+H,Y1,F1,RPAR,IPAR)
C ---- ESTIMATE THE SECOND DERIVATIVE OF THE SOLUTION
DER2=0.0D0
IF (ITOL.EQ.0) THEN
DO 15 I=1,N
SK=ATOLI+RTOLI*ABS(Y(I))
15 DER2=DER2+((F1(I)-F0(I))/SK)**2
ELSE
DO 16 I=1,N
SK=ATOL(I)+RTOL(I)*ABS(Y(I))
16 DER2=DER2+((F1(I)-F0(I))/SK)**2
END IF
DER2=SQRT(DER2)/H
C ---- STEP SIZE IS COMPUTED SUCH THAT
C ---- H**IORD * MAX ( NORM (F0), NORM (DER2)) = 0.01
DER12=MAX(ABS(DER2),SQRT(DNF))
IF (DER12.LE.1.D-15) THEN
H1=MAX(1.0D-6,ABS(H)*1.0D-3)
ELSE
H1=(0.01D0/DER12)**(1.D0/IORD)
END IF
H=MIN(100*ABS(H),H1,HMAX)
HINIT853=SIGN(H,POSNEG)
RETURN
END
C
FUNCTION CONTD8(II,X,CON,ICOMP,ND)
C ----------------------------------------------------------
C THIS FUNCTION CAN BE USED FOR CONINUOUS OUTPUT IN CONNECTION
C WITH THE OUTPUT-SUBROUTINE FOR DOP853. IT PROVIDES AN
C APPROXIMATION TO THE II-TH COMPONENT OF THE SOLUTION AT X.
C ----------------------------------------------------------
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION CON(8*ND),ICOMP(ND)
COMMON /CONDO8/XOLD,H
C ----- COMPUTE PLACE OF II-TH COMPONENT
I=0
DO 5 J=1,ND
IF (ICOMP(J).EQ.II) I=J
5 CONTINUE
IF (I.EQ.0) THEN
WRITE (6,*) ' NO DENSE OUTPUT AVAILABLE FOR COMP.',II
CONTD8=-1
RETURN
END IF
S=(X-XOLD)/H
S1=1.D0-S
CONPAR=CON(I+ND*4)+S*(CON(I+ND*5)+S1*(CON(I+ND*6)+S*CON(I+ND*7)))
CONTD8=CON(I)+S*(CON(I+ND)+S1*(CON(I+ND*2)+S*(CON(I+ND*3)
& +S1*CONPAR)))
RETURN
END
|
